Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $x = \dfrac{10k + 18}{2} \div \dfrac{4k(5k + 9)}{7k} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{10k + 18}{2} \times \dfrac{7k}{4k(5k + 9)} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ (10k + 18) \times 7k } { 2 \times 4k(5k + 9) } $ $ x = \dfrac {7k \times 2(5k + 9)} {2 \times 4k(5k + 9)} $ $ x = \dfrac{14k(5k + 9)}{8k(5k + 9)} $ We can cancel the $5k + 9$ so long as $5k + 9 \neq 0$ Therefore $k \neq -\dfrac{9}{5}$ $x = \dfrac{14k \cancel{(5k + 9})}{8k \cancel{(5k + 9)}} = \dfrac{14k}{8k} = \dfrac{7}{4} $